#7684975

order	512
nilpotency class	2
automorphism	?
homology	Abelianization (H₁) = C₂⁶ · Schur multiplier (H₂) = C₂¹²

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, commutator, Frattini, socle, cosocle	C₂³	C₂⁶
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4815 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 32
transitive degree	μ_t(G) = 256

A minimal faithful permutation representation — 6 generators on 32 points:

(2,5)(6,8)(9,10)(11,14)(12,16)(13,15)(17,18)(19,21)(20,25)(22,26)(23,24)(27,29)(28,30)(31,32);(1,2,4,5)(3,6,7,8)(9,11,13,12)(10,14,15,16)(17,19,23,20)(18,21,24,25)(22,27,31,28)(26,29,32,30);(10,15)(14,16)(18,24)(19,20)(26,32)(27,28);(9,10)(11,14)(12,16)(13,15)(17,20,23,19)(18,25,24,21)(22,28,31,27)(26,30,32,29);(9,12,13,11)(10,16,15,14)(17,21,23,25)(18,19,24,20)(22,30,31,29)(26,28,32,27);(1,3,4,7)(2,6,5,8)(17,22)(18,26)(19,27)(20,28)(21,29)(23,31)(24,32)(25,30)

Polycyclic presentation:

2^9|b^2=gh,d^2=i,e^2=h,f^2=g,b^a=bg,c^a=ch,c^b=ci,d^c=dh,f^e=fi

Conjugacy classes & character table

113 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 7592171);